On ergodicity of some Markov processes

Abstract

We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example, we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. The weak-$^*$ mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer.Comment: Published in at http://dx.doi.org/10.1214/09-AOP513 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene

“…Next they also extended the technique to Markov operators acting on arbitrary Borel measures what allowed them to study operators appearing in the theory of fractal and semi-fractal sets [25]. Recently it was shown that the technique may be used in verifying the ergodic properties of solutions of infinitely dimensional stochastic differential equations [21]. In this paper we study Markov operators corresponding to iterated function systems with some disturbance.…”

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

“…Let u(·, x) be the solution of (1) and v(·, x) be that of (31) with Z replaced by the stochastic convolution S. Since, by Remark 2.3, the embedding H ⊂ H is dense, one can findā ∈ H such that |ā − a| < ε 3 . Next, from Lemma 4.10 we infer the existence of a control c ∈ L 4 (0, T ; H ) ∩ C([0, T ]; H) such that and u c (T, x) =ā where u c (·, x) is defined by (33). Thus, using the decomposition u(·, x) = v(·, x) + S and the definition of u c (·, x) we infer that…”

“…However, several results about the qualitative or long-time behavior of solution of SPDEs driven by Lévy noise have been obtained during the last decade. We refer, among others, to [18,19,33,38,43,45,46,48,49] for results related to the ergodicity, irreducibility and mixing property of several classes of stochastic evolution equations driven by Lévy processes.…”

Criterion on stability for Markov processes applied to a model with jumps

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